Definite Integrals and the Principle of Least Action: Exploring the Connection

In summary, the conversation discusses the derivation of the principle of least action in Goldstein and a theorem in mathematics about definite integrals. The image provided shows that the theorem only works for small variations in the endpoints. The speaker questions why the variation is calculated as L(t2)Δt2 - L(t1)Δt1 instead of L(t2 +Δt2) - L(t1 +Δt1) and suggests that the latter makes more sense. However, it is explained that when the deltas are small, it does not make a difference where the function is evaluated within the interval. The conversation ends with a mention of using a Taylor expansion to further understand this concept.
  • #1
Ben Geoffrey
16
0
This is with regard to my doubt in the derivation of the principle of least of action in Goldstein

Is there any theorem in math about definite integrals like this ∫a+cb+df(x)dx = f(a)c-f(b)d

The relevant portion of the derivation is given in the image.
 

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  • #2
That only works when the variations in the endpoints are small. The integral over a small interval is approx. the function value at a point in the interval times the width of the interval.
 
  • #3
But why is the variation due to ends points L(t2)Δt2 - L(t1)Δt1 rather than L(t2 +Δt2) - L(t1 +Δt1) . Makes more sense if it is L(t2 +Δt2) - L(t1 +Δt1)
 
  • #4
Ben Geoffrey said:
But why is the variation due to ends points L(t2)Δt2 - L(t1)Δt1 rather than L(t2 +Δt2) - L(t1 +Δt1) . Makes more sense if it is L(t2 +Δt2) - L(t1 +Δt1)

If the deltas are small it makes no difference where you evaluate the function within the small interval.

To see this use a Taylor expansion.
 
  • #5
thank you
 

What is the Principle of Least Action?

The Principle of Least Action is a fundamental law in physics that states that the path taken by a physical system between two points in time is the one that minimizes the action, which is a mathematical quantity that takes into account the system's energy and time.

How is the Principle of Least Action used in physics?

The Principle of Least Action is used in physics to predict the motion of objects and systems in a variety of fields, including classical mechanics, quantum mechanics, and electromagnetism. It provides a powerful mathematical framework for understanding the behavior of physical systems.

What are the implications of the Principle of Least Action?

The Principle of Least Action has profound implications for our understanding of the fundamental laws of nature. It suggests that the laws of physics are not arbitrary, but are instead based on the principle of minimizing action. This has led to important developments in our understanding of the universe and has helped shape modern physics theories.

How does the Principle of Least Action relate to other principles in physics?

The Principle of Least Action is closely related to other fundamental principles in physics, such as the principle of least energy and the principle of least time. It is also related to other principles in mathematics and philosophy, such as the principle of least effort and the principle of least resistance.

What are some real-world applications of the Principle of Least Action?

The Principle of Least Action has numerous real-world applications, such as predicting the motion of planets in our solar system, understanding the behavior of particles in particle accelerators, and designing efficient paths for spacecraft. It is also used in fields such as optics, fluid dynamics, and economics.

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